Integration and modern analysis:
Gespeichert in:
| Beteiligte Personen: | , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Boston [u.a.]
Birkhäuser
2009
|
| Schriftenreihe: | Birkhäuser advanced texts
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017662558&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017662558&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XIX, 575 S. graph. Darst. |
| ISBN: | 9780817643065 9780817646561 |
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| 100 | 1 | |a Benedetto, John J. |d 1939- |e Verfasser |0 (DE-588)132081482 |4 aut | |
| 245 | 1 | 0 | |a Integration and modern analysis |c John J. Benedetto ; Wojciech Czaja |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2009 | |
| 300 | |a XIX, 575 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Birkhäuser advanced texts | |
| 650 | 4 | |a Functions of real variables | |
| 650 | 4 | |a Integrals, Generalized | |
| 650 | 4 | |a Integration, Functional | |
| 650 | 4 | |a Mathematical analysis | |
| 650 | 4 | |a Measure theory | |
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Datensatz im Suchindex
| DE-BY-TUM_call_number | 0102 MAT 280f 2009 A 9518 |
|---|---|
| DE-BY-TUM_katkey | 1705494 |
| DE-BY-TUM_location | 01 |
| DE-BY-TUM_media_number | 040010221493 |
| _version_ | 1821933739656085506 |
| adam_text | Table
of
Contents
Preface
хш
Classical Real Variables
................................... 1
1.1
Set theory: a framework
................................. 1
1.2
The topology of
R
...................................... 2
1.3
Classical real variables
.................................. 10
1.3.1
Motivation for the Lebesgue theory
................. 10
1.3.2
Continuous functions
............................. 11
1.3.3
Uniform convergence
.............................. 17
1.3.4
Sets of differentiability
............................ 19
1.4
References for the history of integration theory
............. 25
1.5
Potpourri and
titillation
................................. 26
1.6
Problems
.............................................. 28
Lebesgue Measure and General Measure Theory
.......... 37
2.1
The theory of measure prior to Lebesgue, and preliminaries
. . 37
2.2
The construction of Lebesgue measure on
R
............... 40
2.3
The existence of Lebesgue measure on Rd
.................. 51
2.4
General measure theory
................................. 58
2.5
Approximation theorems for measurable functions
.......... 70
2.6
Potpourri and
titillation
................................. 78
2.7
Problems
.............................................. 83
The Lebesgue Integral
.................................... 95
3.1
Motivation
............................................ 95
3.2
The Lebesgue integral
................................... 99
3.3
The Lebesgue dominated convergence theorem (LDC)
.......105
3.4
The Riemann and Lebesgue integrals
.....................117
3.5
Lebesgue-Stieltjes measure and integral
...................123
3.6
Some fundamental applications
...........................129
3.7
Fubini and Tonelli theorem
..............................132
3.8
Measure zero and sets of uniqueness
......................140
3.8.1
B. Riemann
.....................................140
3.8.2
G. Cantor
.......................................141
3.8.3
D. Menshov
.....................................142
viii
Table of Contents
3.8.4 N.
Bari
and A. Rajchman
.........................142
3.9
Potpourri and
titillation
.................................143
3.10
Problems
..............................................
148
4
The Relationship between Differentiation and Integration
on
R
......................................................165
4.1
Functions of bounded variation and associated measures
.....165
4.2
Decomposition into discrete and continuous parts
...........172
4.3
The Lebesgue differentiation theorem
.....................178
4.4
Fundamental Theorem of Calculus I
......................185
4.5
Absolute continuity and Fundamental Theorem of Calculus II
189
4.6
Absolutely continuous functions
..........................195
4.7
Potpourri and
titillation
.................................201
4.8
Problems
..............................................204
5
Spaces of Measures and the Radon-Nikodym Theorem
... 219
5.1
Signed and complex measures, and the basic decomposition
theorems
..............................................219
5.2
Discrete and continuous, absolutely continuous and singular
measures
..............................................232
5.3
The Vitali-Lebesgue-Radon-Nikodym theorem
.............239
5.4
The relation between set and point functions on
R
..........246
5.5
ĽP(X),
1 <
p
<
oc
.....................................251
5.6
Potpourri and
titillation.................................
261
5.7
Problems
..............................................264
6
Weak Convergence of Measures
...........................277
6.1
Vitali
theorem
.........................................277
6.2
Nikodym and Hahn-Saks theorems
.......................283
6.3
Weak sequential convergence
.............................292
6.4
Dieudonné-Grothendieck
theorem
........................303
6.5
Norm and weak sequential convergence
....................310
6.6
Potpourri and
titillation
.................................316
7
Riesz Representation Theorem
...........................321
7.1
Original Riesz representation theorem
.....................321
7.2
Riesz representation theorem (RRT)
......................323
7.3
Radon measures
........................................336
7.4
Support and the approximation theorem
..................342
7.5
Distribution theory
.....................................345
7.6
Potpourri and
titillation
.................................353
Table
of Contents
ix
8
Lebesgue Differentiation Theorem on Rd
..................359
8.1
Introduction
...........................................359
8.2
Maximal function and Lebesgue differentiation theorem
.....361
8.3
Coverings
.............................................364
8.4
Differentiation of measures
..............................372
8.5
Bounded variation and the divergence theorem
.............380
8.6
Rearrangements and the maximal function theorem
.........384
8.7
Change of variables and surface measure
..................395
8.8
Potpourri and
titillation
.................................401
9
Self-Similar Sets and Fractals
.............................407
9.1
Self-similarity and fractals
...............................407
9.2
Peano curve
...........................................412
9.3
Hausdorff measure
......................................416
9.4
Hausdorff dimension and self-similar sets
..................423
9.5
Lipschitz mappings and generalizations of fractals
..........427
9.6
Potpourri and
titillation
.................................433
A Functional Analysis
.......................................441
A.I Definitions of spaces
....................................441
A.2 Examples
..............................................446
A.3 Separability
............................................450
A.
4
Moore-Smith and
Arzelà-Ascoli
theorems
.................451
A.
5
Uniformly continuous functions
...........................453
A.6 Baire category theorem
..................................454
A.
7
Uniform Boundedness Principle and
Schur
lemma
..........457
A.8 Hahn-Banach theorem
..................................459
A.9 The weak and weak* topologies
..........................466
A.
10
Linear maps
...........................................471
A.ll Embeddings of dual spaces
..............................473
A.12 Hubert spaces
..........................................475
A.13 Operators on Hubert spaces
.............................480
A.14 Potpourri and
titillation
.................................482
В
Fourier Analysis
..........................................487
B.I Fourier transforms
......................................487
B.2 Analytic properties of Fourier transforms
..................491
B.3 Approximate identities
..................................494
B.4 The
L„(R)
theory of Fourier transforms
...................498
B.5 Fourier series
..........................................501
B.6 The Ll(T2n) theory of Fourier series
......................503
B.7 The Stone-Weierstrass theorem
..........................506
B.8 The L2(T2n) theory of Fourier series
......................507
B.9
Haar
measure
..........................................509
B.10 Dual groups and the Fourier analysis of measures
...........514
χ
Table of Contents
В.
11
Radial Fourier transforms
...............................519
B.12 Wiener s Generalized Harmonic Analysis (GHA)
...........522
B.13 Epilogue
..............................................528
References
....................................................529
Subject Index
................................................551
Index of Names
...............................................565
Index of Notation
............................................573
John J. Benedetto and
Wojciech Czają
Integration and Modern Analysis
A paean to twentieth century analysis, this modern text has several
important themes and key features which set it apart from others on the
subject. A major thread throughout is the unifying influence of the
concept of absolute continuity on differentiation and integration. This
leads to fundamental results such as the
Dieudonné-Grothendieck
theorem and other intricate developments dealing with weak
convergence of measures.
Key Features:
•
Fascinating historical commentary interwoven into the exposition;
•
Hundreds of problems from routine to challenging;
•
Broad mathematical perspectives and material, e.g., in harmonic
analysis and probability theory, for independent study projects;
•
Two significant appendices on functional analysis and Fourier
analysis.
Key Topics:
•
In-depth development of measure theory and Lebesgue integration;
•
Comprehensive treatment of connection between differentiation
and integration, as well as complete proofs of state-of-the-art
results;
•
Classical real variables and introduction to the role of Cantor
sets, later placed in the modern setting of self-similarity and fractals;
•
Evolution of the Riesz representation theorem to Radon measures
and distribution theory;
•
Deep results in modern differentiation theory;
___________________
•
Systematic development of weak sequential convergence inspired
by theorems of
Vitali,
Nikodym, and Hahn-Saks;
•
Thorough treatment of rearrangements and maximal functions;
•
The relation between surface measure and Hausforff measure;
•
Complete presentation of Besicovich coverings and differentiation
of measures.
Integration and Modern Analysis will serve advanced undergraduates and
graduate students, as well as professional mathematicians. It may be
used in the classroom or self-study.
|
| any_adam_object | 1 |
| author | Benedetto, John J. 1939- Czaja, Wojciech |
| author_GND | (DE-588)132081482 (DE-588)105191454X |
| author_facet | Benedetto, John J. 1939- Czaja, Wojciech |
| author_role | aut aut |
| author_sort | Benedetto, John J. 1939- |
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| classification_tum | MAT 280f |
| ctrlnum | (OCoLC)320494774 (DE-599)DNB994038771 |
| dewey-full | 515.8 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.8 |
| dewey-search | 515.8 |
| dewey-sort | 3515.8 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV035607364 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T13:38:53Z |
| institution | BVB |
| isbn | 9780817643065 9780817646561 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017662558 |
| oclc_num | 320494774 |
| open_access_boolean | |
| owner | DE-20 DE-91G DE-BY-TUM DE-703 DE-355 DE-BY-UBR DE-188 DE-898 DE-BY-UBR DE-11 DE-739 |
| owner_facet | DE-20 DE-91G DE-BY-TUM DE-703 DE-355 DE-BY-UBR DE-188 DE-898 DE-BY-UBR DE-11 DE-739 |
| physical | XIX, 575 S. graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Birkhäuser advanced texts |
| spellingShingle | Benedetto, John J. 1939- Czaja, Wojciech Integration and modern analysis Functions of real variables Integrals, Generalized Integration, Functional Mathematical analysis Measure theory Integrationstheorie (DE-588)4138369-2 gnd Reelle Funktion (DE-588)4048918-8 gnd Maßtheorie (DE-588)4074626-4 gnd |
| subject_GND | (DE-588)4138369-2 (DE-588)4048918-8 (DE-588)4074626-4 (DE-588)4123623-3 |
| title | Integration and modern analysis |
| title_auth | Integration and modern analysis |
| title_exact_search | Integration and modern analysis |
| title_full | Integration and modern analysis John J. Benedetto ; Wojciech Czaja |
| title_fullStr | Integration and modern analysis John J. Benedetto ; Wojciech Czaja |
| title_full_unstemmed | Integration and modern analysis John J. Benedetto ; Wojciech Czaja |
| title_short | Integration and modern analysis |
| title_sort | integration and modern analysis |
| topic | Functions of real variables Integrals, Generalized Integration, Functional Mathematical analysis Measure theory Integrationstheorie (DE-588)4138369-2 gnd Reelle Funktion (DE-588)4048918-8 gnd Maßtheorie (DE-588)4074626-4 gnd |
| topic_facet | Functions of real variables Integrals, Generalized Integration, Functional Mathematical analysis Measure theory Integrationstheorie Reelle Funktion Maßtheorie Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017662558&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017662558&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT benedettojohnj integrationandmodernanalysis AT czajawojciech integrationandmodernanalysis |
Inhaltsverzeichnis
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Paper/Kapitel scannen lassen
Teilbibliothek Mathematik & Informatik
| Signatur: |
0102 MAT 280f 2009 A 9518 Lageplan |
|---|---|
| Exemplar 1 | Ausleihbar Am Standort |